A Positive Mass Theorem for Spaces with Asymptotic SUSY Compactification

We prove a positive mass theorem for spaces which asymptotically approach a flat Euclidean space times a Calabi-Yau manifold (or any special honolomy manifold except the quaternionic Kähler). This is motivated by the very recent work of Hertog-HorowitzMaeda [HHM]. In general relativity, isolated gravitational systems are modelled by asymptotically flat spacetimes. The spatial slices of such spacetime are then asymptotically flat Riemannian manifolds. That is, Riemannian manifolds (M, g) such that M = M0 ∪ M∞ with M0 compact andM∞ ≃ R −BR(0) for some R > 0 so that in the induced Euclidean coordinates the metric satisfies the asymptotic conditions gij = δij +O(r −τ ), ∂kgij = O(r ), ∂k∂lgij = O(r ). (0.1) Here τ > 0 is the asymptotic order and r is the Euclidean distance to a base point. The total mass (the ADM mass) of the gravitational system can then be defined via a flux integral [ADM], [LP] m(g) = lim R→∞ 1 4ωn ∫ SR (∂igij − ∂jgii) ∗ dxj . (0.2) Here ωn denotes the volume of the n − 1 sphere and SR the Euclidean sphere with radius R centered at the base point. If τ > n−2 2 and n ≥ 2, then m(g) is independent of the asymptotic coordinates xi, and thus is an invariant of the metric. The positive mass theorem [SY1], [SY2], [SY3], [Wi1] says that this total mass is nonnegative provided one has nonnegative local energy density. Theorem 0.1 (Schoen-Yau, Witten). Suppose (M, g) is an asymptotically flat spin manifold of dimension n ≥ 3 and of order τ > n−2 2 . If the scalar curvature R ≥ 0, then m(g) ≥ 0 and m(g) = 0 if and only if M = R. Remark. The scalar curvature R is the local energy density. According to string theory [CHSW], our universe is really ten dimensional, modelled by M×X where X is a Calabi-Yau 3-fold. This is the so called Calabi-Yau compactification, which motivates the spaces we now consider. We consider the complete Riemannian manifolds (M, g) such that M = M0 ∪ M∞ with M0 compact and M∞ ≃ (R k −BR(0)) ×X for some R > 0 and X a compact simply